Finite Root System Research Articles

On symmetric closed subsets of real affine root systems

Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit description of symmetric closed subsets of real affine root systems. At the end, using our results, we study the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them.

Central limit theorems for generalized descents and generalized inversions in finite root systems

Central limit theorems for generalized descents and generalized inversions in finite root systems

Classification of Cocovers in the Double Affine Bruhat Order

We classify cocovers of a given element of the double affine Weyl semigroup $W_{\mathcal{T}}$ with respect to the Bruhat order, specifically when $W_{\mathcal{T}}$ is associated to a finite root system that is irreducible and simply laced. We do so by introducing a graphical representation of the length difference set defined by Muthiah and Orr and identifying the cocovering relations with the corners of those graphs. This new method allows us to prove that there are finitely many cocovers of each $x \in W_{\mathcal{T}}$. Further, we show that the Bruhat intervals in the double affine Bruhat order are finite.

Root systems, affine subspaces, and projections

We tackle several problems related to a finite irreducible crystallographic root system Φ in the real vector space E. In particular, we study the combinatorial structure of the subsets of Φ cut by affine subspaces of E and their projections. As byproducts, we obtain easy algebraic combinatorial proofs of refinements of Oshima's Lemma and of a result by Kostant, a partial result towards the resolution of a problem by Hopkins and Postnikov, and new enumerative results on root systems.

Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

Stability of Stretched Root Systems, Root Posets and Shards

Inspired by the infinite families of finite and affine root systems, we define a "stretching" operation on general crystallographic root systems which, on the level of Coxeter diagrams, replaces a vertex with a path of unlabeled edges. We embed a root system into its stretched versions using a similar operation on individual roots. For a fixed root, we describe the long-term behavior of two associated structures as we lengthen the stretched path: the downset in the root poset and Reading's arrangement of shards. We show that both eventually admit a uniform description, and deduce enumerative consequences: the size of the downset is eventually a polynomial, and the number of shards grows exponentially.

Rank 4 finite-dimensional Nichols algebras of diagonal type in positive characteristic

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. In the present paper, all rank 4 Nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified. Our proof uses the classification of the finite arithmetic root systems of rank 4.

The action of a Coxeter element on an affine root system

The characterization of orbits of roots under the action of a Coxeter element is a fundamental tool in the study of finite root systems and their reflection groups. This paper develops the analogous tool in the affine setting, adding detail and uniformity to a result of Dlab and Ringel.

Higher-order congruence relations on affine moment graphs: The subgeneric case

We study the structure algebra Z of the stable moment graph for the case of the affine root system A1. The structure algebra Z is an algebra over a symmetric algebra and in particular, it is a module over a symmetric algebra. We study this module structure on Z and we construct a basis. By “setting c equal to zero” in Z, we obtain the module Zc=0. This module can be described in terms of the finite root system A1 and we show that it is determined by a set of certain divisibility relations. These relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs, and because of this we call them higher-order congruence relations.

On the Tits cone of a Weyl groupoid

We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality condition which we call the crystallographic property.

Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

We define the formal affine Demazure algebra and a formal affine Hecke algebra associated to a Kac-Moody root system. We prove structure theorems for these algebras, and use them to extend several results and constructions (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson classes, root polynomials and multiplication formulas) that were previously known for finite root systems.

The Weak Order on Weyl Posets

AbstractWe define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

On conjugacy of abstract root bases of root systems of Coxeter groups

We introduce and study a combinatorially defined notion of the root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are extended to abstract root bases, to a larger class of real root systems and, with a short list of (genuine) exceptions, to infinite rank irreducible Coxeter systems.

Parabolic subgroup orbits on finite root systems

Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.

Chip firing on Dynkin diagrams and McKay quivers

This paper establishes new connections between the representation theory of finite groups and sandpile dynamics. Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay–Cartan matrices for finite subgroups G of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay–Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup G. In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.

Maximal Weight Composition Factors for Weyl Modules

AbstractFix an irreducible (finite) root system R and a choice of positive roots. For any algebraically closed field k consider the almost simple, simply connected algebraic group Gk over k with root system k. One associates with any dominant weight λ for R two Gk-modules with highest weight λ, the Weyl module V( λ)k and its simple quotient L(µ)k . Let λ and μ be dominant weights with μ < j such that μ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists k such that L(μ)k is a composition factor of V(j)k , and they exhibit an example in type E8 where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs (λ, μ), and another that relies only on the classiûcation of root systems.

The classification of Nichols algebras over groups with finite root system of rank two

We classify all groups G and all pairs (V,W) of absolutely simple Yetter–Drinfeld modules over G such that the support of V\oplus W generates G , c_{W,V}c_{V,W}\ne\mathrm {id} , and the Nichols algebra of the direct sum of V and W admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter–Drinfeld modules over groups.

Vanishing Cycles and Cartan Eigenvectors

Using the ideas coming from the singularity theory, we study the eigenvectors of the Cartan matrices of finite root systems, and of q-deformations of these matrices

Rank three Nichols algebras of diagonal type over arbitrary fields

We classify all rank three Nichols algebras of diagonal type with a finite root system over fields of arbitrary characteristic. Our proof uses the classification of the finite Weyl groupoids of rank three.

A combinatorial formula expressing periodic R-polynomials

In 1980, Lusztig introduced the periodic Kazhdan–Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac–Moody algebra at the critical level. The periodic Kazhdan–Lusztig polynomials can be computed by using another family of polynomials, called the periodic R-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic R-polynomials in terms of the “doubled” Bruhat graph associated to a finite Weyl group and a finite root system.